Survey: Large Communication Complexity Classes

Since the introduction of communication complexity four decades ago, complexity theorists have generally achieved better results and separations in communication complexity than in normal complexity classes. For instance, it is known that no pair of P, BPP, NP, and PP are equal, while not a single pair of all the corresponding normal classes P, BPP, NP, and PP have been proven different. Fewer results have been shown about larger complexity classes, however, and many of those are not well known. In particular, material on unbounded-error communication complexity classes and on PSPACE is only beginning to materialize. We survey the structure and relations among the large communication classes PSPACE and UPP and related topics, namely space bounds and unbounded error as applied to communication complexity. Along the way, we define a simple flavor of space-bounded communication with public memory, which seems not to have been considered carefully before. We also give a more concise presentation of some commonly-used characterizations of unbounded-error protocols from linear algebra, and examine the separation of UPP and PP.